I want to design a $Decider$ such that for every input accept if and only if Goldbach's conjecture is true and reject all inputs if and only if Goldbach's conjecture has a counter example.
I made a $Decider$ that accept all string if and only if Goldbach's conjecture was right by assuming that Goldbach's conjecture has a $TM_{goldbach}$ simply i check if $w$ is represented a even number or odd number if it represent a odd number i change that to a even by adding a dummy Char for instance and then i pass it to $TM_{goldbach}$ which compute 2 prime number that summation of those make the input length (or the number that input is represented) If $TM_{goldbach}$ halt the $Decider$ accept (Means that $TM_{goldbach}$ find the the equivalent prime numbers).
But this definition accept lots of input even when we have a counter example i know there should be trick that i can reject every string when Goldbach's conjecture was wrong (finding counter example? how?).
Thanks for help.
===edit===
Apparently making such Decider which I've described above leads us to contradiction.
Thanks.